Analysis Of M~x/G/1 Repairable Queuing System In Which The Service Station May Fail In The Idle Periods

Repairable service system will be taken into failure, namely the service station possibly breaks down and might be repaired. Both queuing indices and reliability indices are studied in repairable systems. This paper studies the MX/G/1 repairable queuing system, while the service station may fail and have different failure rates during its busy and idle periods. By using the total probability decomposition, L-transform, generating function and other tools, through the introduction of the"generalized servicing times", the"idle time of system", the"busy time of server"and so on, starting from any initial state the following question are discussed:1) The transient queue length distribution, and the recursion expression of L-transform of the transient queue length distribution is obtained.2) The steady queue length distribution, and the recursion expression of the steady queue length distribution and the probability generating function expression are obtained.3) The probability that the time t is in the"server generalized busy time", and its L-transform and the steady result are obtained .4) The probability of the first type failure and the second type failure, and the unavailability of the service station separately, and its L-transform and the steady result are obtained.5) The expected number of the first type failure and the second type failure during the interval( 0,t ], and its LS-transform and steady breakdown frequency are obtained.